# Introduction to (Formal) Logic (and AI)

Spring 2024 Edition of IFLAI1 (“eye” • “fly” • “one”)

Selmer Bringsjord

## Table of Contents

**Learning logic with patented AI technology.**

with Naveen Sundar G.

∧ KB Foushée ∧ \(\ldots\)

Figure 2: Larry

Figure 3: Lucy

[All artwork (all of which is copyrighted) for the LAMA^{®} paradigm by KB Foushée.]

## Terminology & General Orientation

This course is an advanced, accelerated introduction to deductive
formal logic, with some substantive coverage of
*in*ductive formal logic, in which formalisms
for dealing with uncertainty (e.g., probability theory, and also the
likelihood calculus) are included, and to heterogeneous formal logic
(which allows reasoning over not only textual/linguistic content, but
visual content as well). The course conforms to the
LAMA^{®} paradigm in general, and is specifically
based on *Hyperlogic*^{TM}, which among other things
(all of which are explained and covered) is based on the view that
proofs and arguments are best cast as **hypergraphs**. Since the
present course emphasizes deduction, the course is specifically based
on **hypergraphical natural deduction**. This is the only robust
treatment of formal logic based on this form of deduction, which has
many unique advantages.

The course makes crucial use of patented AI for learning, and also
provides an introduction to AI itself, at least AI of the logicist
variety. In particular, and for example, students are exposed to a
pure and general form of logic programming (so-called **PGLP**) that is
particularly well-suited for pursuing logicist AI; students are also
introduced to the new PGLP programming environment
*Hyperlog*^{®}. In addition, students are
educated about the part of AI known as *automated reasoning*, which in
general consists in AI itself reasoning in both deductive and
inductive fashion.

The last part of the class includes distinctive coverage of some of the great theorems proved by humanity’s greatest logician: Kurt Gödel.

- Terminology
Note that sometimes ‘symbolic’ is used in place of ‘formal.’ This is bad practice, since formal logic includes

*heterogeneous*logics in which not only symbolic, but also pictorial information, figures. Deductive formal logic is a superset of mathematical logic; the latter is the part of deductive formal logic devoted to the mathematical analysis of mathematics itself (which is why some also use the term ‘meta-mathematics’ to denote mathematical logic). Part of the present course is devoted to mathematical logic. The specific phrase used to describe what the student is principally introduced to in this class is:*beginning deductive logic, advanced*(BDLA); hence the title of our textbook (see below). - What Next?
After this class, the student can proceed to the intermediate level in formal deductive and inductive logic, and — with a deeper understanding and better prepared to flourish — to various areas within the

*formal sciences*, which are invariably based on formal logic, at least to a high degree. The formal sciences include e.g. theoretical computer science (e.g., computability theory, complexity theory, rigorous coverage of programming and programming languages), mathematics in all its traditional branches (analysis, geometry, topology, etc), decision theory, (economic) game theory, set theory, probability theory, etc. The class also serves as a stepping stone to further study of AI, of formal/theoretical computer science, and of in-depth logic programming. - What is Formal Logic?
In general, formal logic is the science and engineering of reasoning,

^{1}but even this supremely general description fails to convey the flexibility and enormity of the field. For example, the vast majority of classical mathematics can be deductively derived from a small set of formulae (including, e.g.,**ZFC**set theory, which you’ll be hearing more about, and exploring) expressed in the particular formal logics known as ‘first-order logic’ (FOL = \(\mathscr{L}_1\)) and ‘second-order logic’ (= SOL = \(\mathscr{L}_2\)); you’ll*also*be hearing more about both of these logics as well. In addition, computer science emerged from and is in large part based upon logic (for peerlessly cogent yet non-technical coverage of this emergence, see C Glymour’s*Thinking Things Through*). Logic is indeed the foundation for*all*at once rational-and-rigorous intellectual pursuits. (If you can find a counter-example, i.e. such a pursuit that doesn’t directly and crucially partake of logic, S Bringsjord would be very interested to see it.) - Context: A Research University
You have wisely decided to attend a technical

*research*university, with a faculty-led mission to create new knowledge and technology in collaboration with students. RPI is the oldest such place in the English-speaking world; it may know a thing or two about this mission. The mission drives those who teach you in this class. The last thing we want to do is simply convey to you how*others*present and teach introductory formal logic. As should be obvious by now, Bringsjord thinks he has with others in Motalen invented a better way to define, specify, and present formal logic, and to use computational formal logic for AI, and he is toiling to explain this invention, to explain it to others, and to disseminate the invention in question. The better way in question is denoted by the phrase ‘Logic: A Modern Approach^{®},’ or simply by the abbreviation ‘LAMA^{®},’ pronounced so as to rhyme with ‘llama.’ - A Disclaimer!
Please note that guest lecturers other than NS Govindarajulu and A Bringsjord should not be assumed to have fully affirmed the LAMA

^{®}paradigm. This thus applies specifically to any and all TAs and graduate students. As to what these other thinkers hold in connection with LAMA^{®}, that is an open question. You are free to inquire! - Graduate Teaching Assistants; Further Help
The TA for this Sp 2024 edition of the course is PhD student James Oswald, an expert on computational logic and AI (

`oswalj@rpi.edu`

). Please note again the subsection A Disclaimer!. - Prerequisites
There are no formal prerequisites. However, as said above, this course covers

*formal*logic, and though the course is an introduction, it has also been said above that it’s an*advanced*,*accelerated*one. This implies that — for want of a better phrase — students are expected to have a degree of logico-mathematical maturity, achieved for example through mastery of first-rate coverage of high-school mathematics. You have this maturity and mastery on the assumption that you understood the math you were supposed to learn in order to make it where you are. For example, to get to where you are now, you were supposed to have learned the technique of indirect proof (= proof by contradiction =*reductio ad absurdum*), from either or both of the courses Algebra 2 and Geometry. An example of the list of concepts and techniques you are assumed to be familiar with from high-school geometry can be found in the common-core-connected*Geometry: Common Core*by Bass et al., published by Pearson in 2012. An example of the list of concepts and techniques you are assumed to be familiar with from high-school Algebra 2 can be found in the common-core-connected*Algebra 2: Common Core*by Bellman et al., published by Pearson, 2012. It’s recommended that during the first two weeks of the class, students review their high-school coverage of formal logic. While this material will be covered from scratch in this class, it helps to have at least some command of it from high school, since our pace will be a rapid one. A much more robust treatment of prerequisites and a suitable background for this course is provided in the syllabus.

## Texts/Readings

Students will purchase access to (and in the case of the e-textbook, obtain) the inseperably interconnected trio of

- HyperGrader
^{®}, a comprehensive, patented (Patent No.**US 11,526,779 B2**) AI platform generating, assessing, tracking, and broadcasting (in anonymized form on leaderboards) problems, the tackling of which largely takes place in - the patented HyperSlate
^{®}AI system for (among other things) proof construction in collaboration with AI technology; and - the (copyrighted) e-textbook
*Logic: A Modern Approach; Beginning Deductive Logic via HyperSlate*(LAMA-BDLAHS).^{®}, Advanced

Each member of this trio will be available online after purchase of
the relevant code-carrying envelope from or in the RPI Follett
Bookstore. Full logistics of this purchase, and the content of the
envelope and how to proceed from this content, will be explained in the
first class (and subsequently, in greater details, with walk-throughs
given live). Updates to LAMA-BDLAHS, and additional exercises, will
be provided by listing on relevant LAMA^{®} web
pages upon signing in to HyperGrader^{®} (and
sometimes by email) through the course of the semester. You will need
to manage many electronic files as this course proceeds, and
e-housekeeping and e-orderliness are of paramount importance. You
will specifically need to assemble a library of completed and
partially completed proofs in the cloud so that you can use them as
building blocks in harder proofs; in other words, building up your own
“logical library” will be crucial. This will be your
library in HyperSlate^{®}.

Please note that both HyperGrader^{®} and
HyperSlate^{®} are both patented, copyrighted,
trademarked AI systems; copying and/or reverse engineering or
distributing this software to others is strictly prohibited by law.
You will need to AGREE online (after registration) to a License
Agreement. This agreement will also cover the e-textbook, which is
copyrighted as well, and cannot be copied or distributed in any way,
even in part.

In addition, occasionally papers may be assigned as reading. Two, indeed, were assigned in the syllabus, on the first day of class.

As to AI, it’s strongly recommended that students read the online summary of AI provided by Bringsjord & Govindarajulu, available here.

Finally, slide decks used in class will contain crucial additional
content above and beyond LAMA-BDLA, information posted on
HyperGrader^{®}, and on
HyperSlate^{®}; this additional content will be
available on the web site as the course unfolds through time.

## Syllabus

The version of the course now underway is the Spring **2024** edition,
the syllabus for which is available here. This is a robust, detailed
syllabus, and is required reading — and reading that will pay off,
for sure.

## HyperSlate^{®}

This is the patented intelligent software system used for constructing proofs and arguments in collaboration with AI technology, and is available after registration and sign-in.

## HyperGrader^{®}

This is the patented overarching AI platform (via which
HyperSlate^{®} is accessed) for submitting,
getting assessed, and earning points for proofs and arguments
constructed in HyperSlate^{®}, and is available
after registration and by sign-in.

## LAMA-BDLAHS Textbook

This is the e-textbook for the course, and is obtained after
registration and sign-in, by downloading fromm
HyperGrader^{®}.

## Class-Day Content

**January 8 2024**: General Orientation to the LAMA^{®}Paradigm, Logistics, Mechanics. (S Bringsjord)The syllabus was projected and presented, and discussed, in detail. Please note that the syllabus, in particular, makes clear that students who wish to opt for learning under a different paradigm and a different collection of software systems than that of LAMA

^{®}and its patented AI systems should take*Intro to Logic*in a Fall semester, since the “Stanford paradigm” is in use then.**January 11 2024**: The Immaterial Paradise; Motivating Paradoxes, Puzzles, and \mathcal{R}, Part I (a.k.a. “Why Study Logic?”; S Bringsjord)The many answers to the “Why study formal logic?” question are enumerated, and explained. An avowal of Bringsjord’s immaterialist position on formal logic and human cognition is included. Students for whom

*all*the problems presented are a breeze, can conceivably test out of the course, but one specific requirement in that regard is that such a student must’ve arrived from high school math and/or computer science with full command of**explosion**[viz. ∀ Φ, φ, ψ: if Φ ⊢ φ ∧ ¬ φ, then Φ ⊢ ψ], which is needed at a key juncture in this lecture.**January 15 2024**: MLK Day!No class.

**January 18 2024**: Motivating Paradoxes, Puzzles, Part II (S Bringsjord)We start by briefly taking note of the excellent contrarian

*Wall Street Journal*essay “No, AI Machines Can’t Think” by A. Kessler (Jan 7 2024), and a better-than-the-Turing-Test test mentioned within in it: the Lovelace Test, presented here. And we consider some vintage “AI In The News”: whether AI’s should be conscious, as illogically asserted in a*NY Times*opinion piece (“Without Consciousness, AIs Will Be Sociopaths,” Jan 13 2023). (Illogically? Yes:*Contra*the author, Princeton’s Prof. Graziano, from φ → ψ one cannot validly deduce ¬ φ → ¬ ψ!) Next, after review of more puzzlers, including involving boolean operators and the inference schema*explosion*, we then move to Moriarty’s fiendish ticking-bomb challenge! Can you solve it in time to save the world? Our last puzzler is the Monty Hall Problem (and an even-harder variant thereof), which requires*both*deductive and inductive logic to solve.**January 22 2024**: Whirlwind History of Logic — With Skepticism About The Singularity Derived Therefrom (S Bringsjord)In our time-traveling tour of computational formal logic and AI, we go from Euclid, three centuries BC, to — possibly? — The Singularity in our future, and along the way note that Leibniz, peerless polymath and autodidact, is the inventor of modern formal logic, and that at AI’s DARPA-sponsored dawn in 1956, the automated reasoner LogicTheorist stole the show. The presentation ends with Bringsjordian skepticism about The Singularity in light of (Turing-level) machine impotence in the face of the

*Entscheidungsproblem*.**January 25 2024**: Propositional Calculus I, Emphasis on Our First Oracle (S Bringsjord)This class meeting introduces one of the central tenets of LAMA

^{®}: viz., that making immediate and continuous use of AI in the form of provabiity oracles (here, at the level of the propositional calculus) is indispendable for discovering valid proofs that solve problems. The immediate upshot to note is that proof discovery is a joint and indeed collaborative enterprise that unites superior human intelligence with inferior but nonetheless invaluable machine intelligence in the form of automated reasoning. A full, live, hands-on tutorial regarding the UI of HyperSlate^{®}is given. HyperSlate^{®}’s oracles are given.- A video showing the use of the PC provability oracle in
HyperSlate
^{®}solving the “NYS 3” logic problem is available here.

- A video showing the use of the PC provability oracle in
HyperSlate
**January 29 2024**: Propositional Calculus II (S Bringsjord)The simple but cornerstone inference rule of

*modus ponens*(a.k.a.*conditional elimination*, in the natural-reasoning tradition we are following) is introduced, and claimed to be an element of the immaterial universe of of formal logic. This class meeting also introduces the powerful and — at least when it comes to precise, verifiable, valid reasoning — ubiquitous proof technique*proof by cases*, which in HyperSlate^{®}corresponds to*disjunction elimination*. Tutorials are provided via real-time use of HyperSlate^{®}on key exercises.**February 1 2024**: Propositional Calculus III:*Reductio ad Absurdum*Proofs (S Bringsjord)A crucial class, in which, harkening back to the astounding Euclid, we look at his still-edifying indirect proof that there must be infinitely many prime numbers, visit Algebra 2 (and also Geometry) from high school and note that in decent textbooks at even this level there is in fact excellent coverage of indirect proof, and then see how

*reductio ad absurdum*is formalized in HyperSlate^{®}.**February 5 2024**: Pure General Logic Programming (PGLP), and Hyperlog, Part 1Every proof in HyperSlate

^{®}, from the tiny to the elaborate, can be viewed as a Pure General Logic Programming (PGLP) program, and starting from an explanation of this fact (with some look-ahead discussion of the Turing-completeness of the first three elementary deductive logics covered in this class), PGLP, invented by S Bringsjord under support from both AFOSR and France’s ANR, is presented, in its historical context. This lecture also serves as a first introduction to the novel programming language and environment Hyperlog^{®}; this new language is an instance of PGLP available in HyperSlate^{®}, and is integrated with Clojure. In the second part of class, we review, in a help session, in HyperSlate^{®}, the propositional calculus = and the problems students are facing so far.**February 8 2024**: Darwin’s Mistake+; The Pure Predicate Calculus = \(\mathscr{L}_0\);*Toward*Quantification (Selmer Bringsjord)We begin by noting that formal logic provides a way of seeing,

*contra*Darwin (in*Descent of Man*), that*H. sapiens sapiens*is qualitatively distinct, intellectually speaking, from nonhuman animals, in light of the former’s ability to reason (1) with abstract relations and functions, (2) with quantifiers, (3) recursively, and (4) over and with infinite structures. The pure predicate calculus = zeroth-order logic = \(\mathscr{L}_0\) is introduced specifically in connection with (1), which, in contrast to the weaker propositional calculus = \(\mathscr{L}_{pc}\), it regiments and allows. Presentation and discussion is rendered concrete by use of HyperSlate^{®}.- A full video lecture (earlier version), replete with explanation of the two new inference rules that distinguish zero-order logic from the mere propositional calculus, is available here.

**February 12–20 2024**: Test 1Test 1 consists of six problems, three \(\ge \mathscr{L}_0\) problems of ascending difficulty (AI-generated), three human-authored \(\ge \mathscr{L}_{pc}\) problems of ascending difficulty.

For some help, some solution videos for some

*earlier-year*problems (in earlier, less-powerful versions of HyperSlate^{®}) are given immediately below, but note that Selmer is here solving his*own*fourth problem, personalized just for him. One will learn a lot from watching these videos (and stopping and studying them along the way).- The solution to the 2019 problem of showing \(\vdash_{L_{pc}} (A \rightarrow B) \rightarrow (B \rightarrow C) \rightarrow (A \rightarrow C)\) is available here.
- Phase 1 of the solution to EmmaHelpedToo, in which a proof-by-cases
proof plan is created, is available here. EmmaHelpedToo was first
presented in the “motivation” part (Part I) of the class; we have
seen in class a valid informal proof that cracks this problem, and
that prof is mapped into a formal correlate in
HyperSlate
^{®}, and expressed specifically as the proof-by-cases proof plan.

- The solution to one of Selmer’s personalized problems (from a few years back), which is rather “creative” on the part of the AI that produced it, is available here. As shown in the video, surpisingly, three different routes are available for derving the GOAL from a subset of the GIVENs.

**February 15 2024**: Quantification; FOL I; “Proving” God’s Existence (S Bringsjord)Here begins coverage of \(\mathscr{L}_1\), and we thereby begin Part III of the course. We start by noting the need for quantification in some very simple arguments, then explicitly introduce the two simplest inference rules/schemata for quantifiers in \(\mathscr{L}_1\). This meeting starts our coverage of the long-inseparable link between formal logic and purported proofs of God’s existence. We start with two such “proofs,” neither of which is intended to be taken seriously, but both of which have significant pedagogical value. Along the way we note that a serious series of purported proofs of God’s existence have been given by Anselm, Descartes, Leibniz, and — in the 20th century — by the greatest logician so far: Gödel. Analysis of Gödel’s proof will need to wait for a later date; it’s by the way analyzed in a chapter in Bringsjord’s forthcoming

*Gödel’s Great Theorems*, from Oxford University Press.- A full (but in-the-past) video lecture, replete with explanation of the first two new inference rules/schemata for hypergraphical first-order logic, is available here.

**Feb 20 2024**: FOL II (including`universal intro`

) (S Bringsjord)After the usual pointer to an AI-in-the-News issue, the new inference rule/schema

`universal intro`

is introduced, we move to HyperSlate^{®}in order to bring to life a simple but esemplastic example. We end by looking at some excercises that require putting it into action to reach some theorems (including three regarding the relative smartness of different Scandinavian countries).**February 22 2024**: FOL III (existential elim); Logic, AI, Games, and the (Powerful!) Human Mind (S Bringsjord)We first visit a ranking of the “hardness” of problems for humans and AIs based on formal logic. Next, the fourth and final new inference schema for FOL,

`existential elim`

, is covered. In addition, after setting context by reviewing AI’s unimpressive victories in checkers, chess,*Jeopardy!*, and Go, some remarks are made on the nature of grid-based logic-puzzle games; specifically, it’s pointed out that when such games are played without obtaining proofs for the moves made by the player, the efficacy of such games, learning-wise, is questionable. We also discuss Motalen’s latest game: LogiNim.**February 26 2024**: Exhortation; Truth Trees; Measuring Human/Machine Intelligence; and FOL IV (S Bringsjord)Truth trees are introduced, their superiority over truth tables is explained, and it’s shown that HyperSlate

^{®}easily allows for the construction, testing, and*certification*of truth trees — in the very same environment used for the construction and testing of formal hypergraphical proofs. A challenge to Singularity-affirming thinkers, in the form of a very humble set of problems involving*numerical quantification*, is issued. This challenge comes after a high-altitude view of the basic machinery of this quantification, directly related to a relevant section in the LAMA-BDLA textbook. We end by looking at an artificial multi-agent, embodied in RAIR-Lab robots, solving a challenge that involves robust quantification. This challenge is planned to be one the new RAIR-Lab robot PERI.2 will be able to eventually handle!**February 29 2024**: The Liar; Russell’s Paradox; (Toward, only) Thoraf’s Paradox (S Bringsjord)We introduce our first paradox, perhaps the primogenitor of all paradoxes: The Liar Paradox, or just simply: The Liar. Next, we proceed to Russell’s Paradox, including the Norwegian barberesque version of it, which will soon be a Required Problem on HyperGrader

^{®}. In fact, we mention two forthcoming problems in HyperSlate^{®},`ChimericalBarber`

, and the problem that asks you to create the proof that Russell mailed Frege, and left a large part of the latter’s work in shambles as a result (`RusselsLetter2Frege`

). Finally, we consider two problems that pave the way toward (Thoraf) Skolem’s Paradox (which we don’t directly engage).**Spring Break**! March 2–10 2024

## Tutorials (expands as semester unfolds)

- Getting Registered with HyperGrader
^{®}/HyperSlate^{®}A vintage-but-still-helpful simple tutorial is available here, in which someone registers into the HyperGrader

^{®}platform, and then activates HyperSlate^{®}to prove that switching the order of conjuncts is provable in the propositional calculus = \(\mathscr{L}_{PC}\). This proof wins a trophy and earns a leaderboard spot for the Exercise`switching_conjuncts_fine`

. - Switching Disjuncts is Fine (under 90 sec)
- Just click here.

- Getting Started with HyperSlate
^{®}: Propositional CalculusIn this tutorial, available here, S Bringsjord starts from scratch with a blank workspace and uses HyperSlate

^{®}and the PC provability oracle available therein to show that the atomic proposition \(h\) can be deduced from \(\{\neg \neg c, c \rightarrow a, \neg a \vee b, b \rightarrow d, \neg (d \vee e)\}\). The main purpose of this video is coverage of the intuitive interface and basic moves therein.

## Pop Problems

These problems are timed and released in
HyperGrader^{®} in the absence of any preceding
announcement that they are coming. Please see the syllabus for more
information.

## Homeworks

Homework consists of solving *all* **Required** problems listed on
HyperGrader^{®}’s web pages. (Non-required
problems are clearly marked as such, e.g. as **Bonus Problems**.) All
solutions are created in their final form in
HyperSlate^{®}.
HyperGrader^{®} for interactive use via its
underlying AI technology opens for its Spring 2023 stint on or about
Jan 26 2023, and an orientation/introduction to the system is given in
class that day. Note that homeworks cannot be done without access to,
and sustained and continuous use of,
HyperSlate^{®}, access via the AI platform
HyperGrader^{®}.

## Tests

There are three tests, each presented and taken online and outside of
class via HyperGrader^{®}. Please see the
syllabus for their release dates.

## Footnotes:

^{1}

Warning: Increasingly, the term ‘reasoning’ is used by some who don’t really do anything related to reasoning, as traditionally understood, to nonetheless label what they do. Fortunately, it’s easy to verify that some reasoning is that which is covered by formal logic: If the reasoning is explicit; links declarative statements or formulae together via explicit, abstract reasoning schemata or rules of inference (giving rise to at least explicit arguments, and often proofs); is surveyable and inspectable, and ultimately machine-checkable; then the reasoning in question is what formal logic is the science and engineering of. (An immediate consequence of the characteristics just listed is that AIs based on artificial neural networks don’t reason, ever.) In order to characterize /in/formal logic, one can remove from the previous sentence the requirements that the links must conform to explicit reasoning schemas or rules of inference, and machine-checkability. It follows that so-called informal logic would revolve around arguments, but not proofs. An excellent overview of informal logic, which will be completely ignored in this class, is provided in “Informal Logic” in the Stanford Encyclopedia of Philosophy. In this article, it’s made clear that, yes, informal logic concentrates on the nature and uses of argument.